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Theorem cnvkxpk 4276
 Description: The converse of a Kuratowski cross product. (Contributed by SF, 13-Jan-2015.)
Assertion
Ref Expression
cnvkxpk k(A ×k B) = (B ×k A)

Proof of Theorem cnvkxpk
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvkssvvk 4275 . 2 k(A ×k B) (V ×k V)
2 xpkssvvk 4210 . 2 (B ×k A) (V ×k V)
3 ancom 437 . . 3 ((y A x B) ↔ (x B y A))
4 vex 2862 . . . . 5 x V
5 vex 2862 . . . . 5 y V
64, 5opkelcnvk 4250 . . . 4 (⟪x, y k(A ×k B) ↔ ⟪y, x (A ×k B))
75, 4opkelxpk 4248 . . . 4 (⟪y, x (A ×k B) ↔ (y A x B))
86, 7bitri 240 . . 3 (⟪x, y k(A ×k B) ↔ (y A x B))
94, 5opkelxpk 4248 . . 3 (⟪x, y (B ×k A) ↔ (x B y A))
103, 8, 93bitr4i 268 . 2 (⟪x, y k(A ×k B) ↔ ⟪x, y (B ×k A))
111, 2, 10eqrelkriiv 4213 1 k(A ×k B) = (B ×k A)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ⟪copk 4057   ×k cxpk 4174  ◡kccnvk 4175 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-opk 4058  df-xpk 4185  df-cnvk 4186 This theorem is referenced by:  xpkexg  4288
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